How many TCR clonotypes does a body maintain?

Abstract

We consider the lifetime of a T cell clonotype, the set of T cells with the same T cell receptor, from its thymic origin to its extinction in a multiclonal repertoire. Using published estimates of total cell numbers and thymic production rates, we calculate the mean number of cells per TCR clonotype, and the total number of clonotypes, in mice and humans. When there is little peripheral division, as in a mouse, the number of cells per clonotype is small and governed by the number of cells with identical TCR that exit the thymus. In humans, peripheral division is important and a clonotype may survive for decades, during which it expands to comprise many cells. We therefore devise and analyse a computational model of homeostasis of a multiclonal population. Each T cell in the model competes for self pMHC stimuli, cells of any one clonotype only recognising a small fraction of the many subsets of stimuli. A constant mean total number of cells is maintained by a balance between cell division and death, and a stable number of clonotypes by a balance between thymic production of new clonotypes and extinction of existing ones. The number of distinct clonotypes in a human body may be smaller than the total number of naive T cells by only one order of magnitude.

Keywords: Clonal repertoire; Competition; Extinction; Homeostasis; Stochastic modelling; T cells.

Graphical abstract

Fig. 1

Representation of connections between N=30 T cell clonotypes and M=40 self pMHC subsets. An arrow connecting a red ball to a green ball indicates that the self pMHC stimulates division of cells in the T cell clonotype. The connections between clonotypes and self pMHCs are assigned randomly with probability p=0.1. That is, each entry of the matrix A, independently, is equal to 1 with probability p and equal to 0 with probability $1-p$. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Fig. 2

Predicted steady-state number of distinct clonotypes. The value of N^⁎ is the product of the rate of production of new clonotypes, θ, and the mean lifetime of a clonotype, (16). The dotted lines are (7), valid in the weak-thymus limit. The parameter $\alpha ={\displaystyle \frac{n_{\theta }\theta }{\gamma M}}$ measures the strength of thymic production relative to peripheral division. Three values of $n_{\theta }$ are shown. We use $\gamma M/\mu ={10}^{11}$ cells, approximately equal to the number of naive CD4⁺ T cells in a human.

Fig. 3

A numerical solution of the competition model without thymic production. The left panel shows the total number of T cells, as a function of time. (The total number of cells continues to fluctuate about the same mean for times later than shown.) The right panel shows the number of surviving clonotypes, N(t), with a logarithmic time scale. The dotted line is (14). One time unit is the mean lifetime of a T cell in the absence of division stimulus, taken to be one year in a human body. The parameters are $\mu =1.0$, $\gamma =10$, M=1000; T cell clonotype-pMHC connections were assigned randomly with probability p=0.1 at the beginning of the run. The initial conditions are $N\left(0\right)=200$ and $n_i\left(0\right)=25$ for each i.

Fig. 4

Diagrammatic representation of the competition model. Each green ball represents a set of T cells; each red ball, a set of self pMHC complexes. An arrow (link) connecting a red ball to a green indicates that T cells of the clonotype recognise the self pMHC subset. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Fig. 5

A numerical solution of the exact clonal competition model without thymic input. In the top panel, the average number of self pMHCs recognised by a clonotype, $\overline{\varphi }\left(t\right)$, and the average number of clonotypes recognising a pMHC, $\overline{C}\left(t\right)$, are shown as a function of time. Due to extinction of clonotypes, $\overline{\varphi }\left(t\right)$ increases and $\overline{C}\left(t\right)$ decreases. The middle panel shows histograms of the 2000 values of $\left|C_q\right|$, at the start and end of the realisation. The bottom panel shows the histograms of values of ϕ at the start (1000 clonotypes) and end (168 clonotypes). T cell clonotype-pMHC connections are assigned randomly with probability p=0.05. The remaining parameter values are $\mu =1.0$, $\gamma =10$, M=2000. The initial conditions are $N\left(0\right)=1000$ and $n_i\left(0\right)=10\forall i$.

Fig. 6

Competition is more important than chance in determining which clonotypes survive. Of the initial 1000 clonotypes, on average only 50 survive. Ten independent realisations were carried out with the same connection matrix A; the fates of clonotypes i=390 to i=610 (horizontal axes) are illustrated. The upper panel shows the values of ϕ_i for the subset of clonotypes. (Each i has the same value of ϕ_i in each realisation.) In the lower panel, the vertical axis is the realisation number. A blue rectangle at position i in realisation j indicates that the clonotype has survived to $t={10}^4$ in that realisation. Many clonotypes never survive; some usually do. The parameter values are $\gamma =10$, M=1000, p=0.1. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Fig. 7

Scaling of the dynamics with the size of the self pMHC environment. Different numerical runs, with M=1600, M=3200 and M=6400, without thymic input. In each case, pM=100, $\mu =1$, $N\left(0\right)=\frac{1}{4}M$ and $\gamma =10$. In the vertical axis on the right, the number of surviving clonotypes, as a function of time, is multiplied by p. The product pM is the mean number of self pMHC recognised by a single TCR clonotype; the product pN is the mean number of clonotypes that recognise a single self pMHC.

Fig. 8

Input of new clonotypes from the thymus, while barely affecting the total number of cells, determines the late-time number of surviving clonotypes. For comparison with adult human homeostasis, time is measured in years. The thymic output rates are $\theta =0.1$ year⁻¹ (blue), $\theta =1$ year⁻¹ (green), $\theta =10$ year⁻¹ (red), $\theta =100$ year⁻¹ (black) and $\theta =400$ year⁻¹ (cyan). The remaining parameter values are M=4000, pM=100, $\gamma =10$ year⁻¹, $\mu =1$ year⁻¹, $n_{\theta }=4$ in all cases. The histograms on the right show, for $\theta =10$ year⁻¹, the distribution of survival times of clonotypes from the thymus and the distribution of n_i(t) for surviving clonotypes. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Fig. 9

Total number of cells, and total number of surviving clonotypes, as a function of time. Stimulus rates of the self pMHC, γ_q, are drawn from a log-normal distribution. The values of σ are the ratio of the standard deviation to the mean value of 10 year⁻¹. In the case $\sigma =1$, the standard deviation is equal to the mean. The remaining parameter values are M=4000, pM=100, $\theta =400$ year⁻¹, $\mu =1$ year⁻¹ and $n_{\theta }=4$.